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There is a variety of (good) sources of light (see previous page) but lasers have nice properties, including monochromaticity, perfect coherence, high intensity and variable polarization. In what follows we will make good use of all these features. Outline ☼ Preliminaries Elecromagnetic Radiation Field matter interaction The electric dipole Hamiltonian ☼ Energy-domain spectroscopy (monochromaticity) Electric dipole selection rules Atomic spectra Molecular spectra ☼ Time-domain spectroscopy (coherence) Forming superposition states Electronic wavepackets Vibrational wavepackets Rotational wavepackets (intensity, polarization) ☼ Optical control Using the coherence property Using the intensity property Using the polarization property

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A (very!) qualitative picture of light-matter interaction: Imagine a spring that we drive by tapping on the weight at the natural spring frequency (“on resonance”). Curve (a) represents the spring’s oscillation, curve (b) is the applied force that is required to make it oscillate (“absorption”) and curve (c) is the applied force that is required to stop it from oscillating (“stimulated emission”) Consider next the response of an atom or a molecule. The resonant driving force polarizes (distorts) the system by mixing the initial with an excited state wavefunction. In this cartoon we imagine polarizing the 1s orbital of the H-atom by applying a field at resonance with the 1s  2p transition. As time progresses from t 1 to t 4, the 2p character is increased, then the superposition returns to the initial composition. Subsequently it will oscillate. Fig. 1 Fig. 2

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I. Preliminaries : I-1. A (Very) Brief Introduction to Classical Electrodynamics The use of classical mechanics for description of the field and quantum mechanics for description of the material system is a good approximation in all cases we will consider, since the number of photons in the field will be large throughout the intensity range of interest: Maxwell’s equations provide a complete description of the electric and magnetic fields associated with electromagnetic waves, but to construct a Hamiltonian we need a potential. Introducing a vector potential and a scalar potential we are left with an over-determined problem and hence freedom to choose a constraint – a gauge – within which the wave is uniquely described. The Coulomb gauge (  =0) leads to a plane-wave description of the electric and magnetic fields: Light sourceIntensity (Wcm -2 )photon density (cm -3 ) Mercury lamp Continuous laser~ Moderate pulsed laser~

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Since,, where ê is the polarization direction of the vector potential. (1) (2) (3) (4) (5) { We choose the time origin such that A 0 is purely imaginary,

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It follows that We define electric and magnetic field amplitudes as, in terms of which, Fig. 3 (6) (7)

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with components, Although the Lorentz force is not derived from a potential energy, a Lagrangian for the problem can be found, and can be seen to yield Newton’s equations of motion corresponding to the force (8) by substitution in Lagrange’s equations, Given a Lagrangian, we can calculate the conjugate momenta of the Cartesian coordinates of the particle, e.g., with which the classical Hamiltonian is, I-2. Hamiltonian for a Charged Particle Interacting with a Radiation Field The Lorentz force on a particle of charge q moving at velocity is, (8) (9) (11) (12) (13) (10)

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Relating the velocity to the momentum through, we re-express the Hamiltonian as, Here we consider a collection of mutually interacting charges (e.g., a molecule). Under field- free conditions, the Hamiltonian describing the system is, whereas in the presence of the field it can be written as, (14) (15) (16) (17)

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Expanding the square one finds, small(18) (19) With the quadratic term in A assumed small as compared to the linear term, the Hamiltonian is cast in the anticipated form, where,

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Given the classical Hamiltonian in the form (19), we are in position to substitute quantum operators for the classical momenta, whereby, Here, [i.e., ], but since we are working in the Coulomb gauge and one finds, (20) (21)

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I-3. The Electric Dipole Approximation Recall, where is the wavelength of the electromagnetic field in Fig. 3. In the UV regime  ~ nm, in the visible ~ nm and in the IR l=1  m-1 mm, whereas the molecular system is of sub nm size. We thus expand the exponent in V(t) as, where is the center of mass of the molecule. In the limit, we neglect all terms but the first. Setting we have, The approximation we derived (the electric dipole approximation) applies to the limit. It holds for UV and visible light but not for X-ray radiation, for instance. Higher order terms need be retained also in many other instances, e.g., when the electric dipole interaction is forbidden (and hence the zero order term in the expansion vanishes when matrix elements of the interaction are taken) or when one is interested in propagation effects. Inserting the plane-wave solution for the vector potential we find, (22)

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Inserting, we express the electric dipole Hamiltonian as, (the equivalence of this and the more familiar form can be shown by a gauge transformation) But what does it mean ? Consider an electron interacting by Coulomb forces with an atomic core and subject to a radiation field. Within the electric dipole approximation, Ehrenfest’s theorem gives, (23) (24) (25)

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Eliminating, we find, We see that (within the electric dipole approximation) the center of the wavepacket associated with the electron evolves like a particle of mass m and charge q that is subject to the central force of the core and a spatially uniform electric field. I-4. Transition Rates and the Golden Rule Approximation: Consider a molecular system that evolves subject to a Hamiltonian of the form, Denoting the solutions of the time-dependent Schroedinger equation corresponding to the stationary Hamiltonian H 0 by  j (t), we have, where  j are the eigenstates of H 0, (26) (27) (28) (29)

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The system wavefunction can now be expanded in the complete basis of stationary eigenstates as, Substituting  (t) into the Schroedinger equation we have, and from which or (30) (31) (32) (33) (34) (35)

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Using the orthonormality of the expansion eigenstates we have, or Formally Eq. (37) can be solved to give, The probability of transition from an initial state to a final state of the zero order Hamiltonian is then, Note that the interaction appears (here and more generally) in the form of matrix elements. In this form it can be readily transformed as, (36) (37) (38) (39) (40)

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(41) (42) (43) It follows that the matrix elements of V(t) can be written, A nonperturbative solution to the set of coupled differential equations (37) can be obtained numerically, by propagating the set of coupled equations for sufficiently long time for the time- dependent interaction to have decayed. Such solution will be needed for most applications to be discussed later in the afternoon. In cases where the coupling is small with respect to H 0 (as, for instance, energy-domain spectroscopy), first order perturbation theory is applicable. In this framework it is assumed

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that the amplitude of the initially prepared state remains constant throughout, Using Eq. (44) in Eq. (38) we have, Eq. (45) is general but we will focus on the case where the perturbation is the dipole interaction of the material system with a laser pulse, where E 0 (t) is the pulse envelope. Here the amplitude of interest is proportional to the (incomplete) Fourier transform of the pulse envelope, Our interest is generally in the form of the amplitudes after turn off of the pulse. In that case the amplitude is, up to a constant, the (complete) Fourier transform of the pulse envelope. Assuming a Gaussian profile in time and energy, (44) (45) (46) (47)

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With Eq. (48), the amplitudes in Eq. (47) take the form, The Rotating Wave Approximation amounts to neglect of the second term in the brackets as compared to the first one. In the combined limits where  → ∞ and E m → 0 such that the pulse energy is constant (relevant to high-resolution spectroscopy), the Gaussian reduces to a  -function of energy and, This is the Golden Rule limit. (48) (49) (50) (51) pulse energy = E p { small

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II. Energy Domain Spectroscopy II-1. Atomic Spectra Atomic spectra contain a good deal of physics and have had an important role in the development of quantum mechanics and the history of science. Electric dipole selection rules follow from the requirement that the integrand in will be symmetric under all symmetry operations of the point group (here the full rotation group, R 3 ). Under inversion, r → -r, an atomic wavefunction with angular momentum l has parity (-1) l. The parity of the integrand is therefore, Hence, the only allowed electric-dipole transitions are those involving a change in parity. The atomic orbitals are angular momentum eigenstates and span the irreducible representations and. The electric dipole moment operator behaves as a translation and is therefore described as a superposition of spherical harmonic with angular momentum 1, (52)

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Therefore, from which follows the selection rule, To develop selection rules for m l, the magnetic quantum number, we note that the photon has an intrinsic helicity – the projection of its angular momentum on the quantization axis. For circularly polarized field, it is conventional to define the quantization axis as the line of flight, with which absorption of left-circularly polarized light results in  m l =1 and emission  m l =-1, vice versa for right circularly polarized light. In the general case Fig. 4: Photon absorption can result in either  l=1 or  l=-1.The basis for this selection rule is conservation of angular momentum and the fact that the photon has helicity. l l 1 l+1 l l 1 (53) (54) (55)

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the selection rule is, Algebraically the rotational selection rules are easy to derive. We express the atomic wavefunction as a product of a radial function by an eigenstate of the angular momentum operator, finding for the angular part of the dipole operator matrix element, For the first 3-j symbol on the right to be non-zero m f need be m i +q, where q takes values 0 and ±1. For the second to be nonzero l f need be l i ±1. Recall we invoked the electric dipole approximation – electric dipole forbidden transitions may well be, e.g., magnetic dipole or electric quadrupole allowed. (56) (57) (58)

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II-2. Molecular Spectra II-2-A. Pure Rotational Spectra We assume that (owing to the time-scale disparity) the wavefunction can be separated into an electronic, a vibrational and a rotational part, and hence the electric dipole matrix element between ro-vibronic states takes the form of a matrix element of the permanent electric dipole in a given vibronic state between rotational functions. Linear molecules rotate about two angles and are eigenfunctions of two operators, the material angular momentum squared,, and the space-fixed z-projection of the operator J z, corresponding to two conserved quantum numbers - J and M J. Fig. 5: The Euler angles relate the space- and body-fixed coordinate systems, where  is the polar angle between the space- and body-fixed z- axes, and  and  are the azimuthal angles of rotation about the space- and body-fixed z-axes, respectively.

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where  nv is the permanent dipole moment operator in state n,v. It follows that: Only polar molecules can have a pure rotational spectrum If the dipole operator is expressed in space-fixed Cartesian coordinates,  = x, y, z, Eq. (59) is given as where the Cartesian components of the permanent dipole moment are readily expressed in terms of the spherical harmonics as, (59) (60) (61) (62)

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The selection rules are therefore determined by the same integral we discussed in the context of atoms, which has a component that spans the totally symmetric irreducible representation only if i.e., if J’=J, J±1 (excluding J’=J= 0) and M J +q-M J ’= 0. Observable transitions therefore obey, Symmetric top rotational functions are eigenstates of the total angular momentum squared operator and its space- and body-z projection operators with eigenvalues J, M and K, E J =BJ(J+1) (63) (64) (65) (66)

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Since, however, the permanent dipole operator must be independent of  it is described by a combination of, which are proportional to the spherical harmonics. It follows that The corresponding eigenvalues are where the rotational constants are given in energy unites, A = ħ 2 /2I aa etc, centrifugal distortion is neglected, and the I kk, k=a,b,c, are principal moments of inertia, evaluated at the equilibrium configuration. The pure rotational spectra of both linear and symmetric rotors therefore consists of series of lines separated by 2B (in the absence of centrifugal distortion). E JK =CJ(J+1)+(A-C)K 2 (prolate symmetric top) =AJ(J+1)+(C-A)K 2 (oblate symmetric top) (67) (68) (69)

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The rotational eigenstates of asymmetric tops are not analytically expressible in general, and are typically written as superpositions of symmetric top functions with coefficients determined by diagonalization of the field-free rotational Hamiltonian. The electric dipole matrix element is thus a multiple sum of integrals of the form, [c.f. Eqs. (58) and (64)]. 2E J  =(A+C)J(J+1)+(A-C)E J  (  ) (70)

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II-2-B. Vibrational Spectra Assuming validity of the Born Oppenheimer approximation, the transition matrix element of relevance is a vibrational matrix element of the dipole moment in electronic state n, For a diatomic molecule the dependence of on the vibrational coordinate is often expressed as a power series in the displacement from equilibrium x, To show a vibrational spectrum a diatomic molecule must have a dipole moment that varies with extension For small displacements from equilibrium, (71) (72) (73) (74)

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In the harmonic limit this approximation leads to a “weak selection rule”: where m is the reduced mass,  is the vibrational frequency and a, a + are the annihilation and creation operators, (75) (76) (77) (78)

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where is a Franck-Condon factor. II-2-C. Electronic Spectra The classical basis of the Franck-Condon principle, wherein the electronic system undergoes a vertical transition terminating at the turning point of the excited state while the nuclei neither move nor accelerate: The quantum mechanical version, where the molecule makes a transition from the initial vibrational level into the level with a vibrational wavefunction that best overlaps the initial function: (79) Fig. 6

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III-1. Forming Superposition States In the limit of High resolution spectroscopy: The bandwidth is narrow with respect to the level spacing of the spectrum considered (i.e., the pulse is long with respect to the system timescale) The interaction is small compared to the field-free Hamiltonian where, in the general case, the set of coupled equations is solved numerically during the pulse and propagated analytically subsequently. (80)

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Also simple is the (strongly quantum mechanical) case of a Two-Level System, After the end of the pulse the expectation value of the position operator is, where the interference term oscillates at 2  /  12 (and likewise observables that probe the probability density). In practice this is a “Young 2-slits” experiment: E2E2 E1E1 E2E2 E1E1 (81) (82) Fig. 7

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After the pulse turn-off, the wavepacket evolution is determined solely by the energy phases (and hence conveys information about the excited manifold Hamiltonian), To understand the structure of the short- and long-time evolution, we consider quantum systems that exhibit regular periodic motion in the classical limit and expand the eigenvalue E j about as a power series in j-j *, where j * is the center of the wavepacket in quantum number space, Here,  j+1,j approaches the inverse period of the classical motion,  cl =2  /T cl, in the ħ → 0 limit (where the spectrum is quasi-equidistant) but in general, Intensive research of the broad superposition case was initially motivated by the thought that wavepackets should buy us a route to the classical limit. (83) (84)

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where energy is measured with respect to the wavepacket center energy. From Eq. (86), we have that (in nonlinear quantum systems that execute regular periodic motion in the classical limit): If the wavepacket is spatially localized (i.e., its spatial dimension  x is small as compared to the dimension L of the classical orbit corresponding to the wavepacket center energy) at t = 0, its evolution at times t « T rev will mimic the evolution of a classical particle. At times t non-negligible with respect to T rev, the nonlinearity of the spectrum will become observable and the wavepacket will start dephasing at a rate that scales as  j 2, where  j is the width of the distribution in quantum number space. At multiples of T rev, the t = 0 wavepacket will be precisely reconstructed and for a while (for a period  t « T rev ) will again evolve nearly classically – more so the smaller  x/L. (85) (86) With the definition the wavepacket of Eq. (83) takes the form,

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As a numerical approach, the basis set expansion [see Eqs. (80)] is simple in concept and essentially universal, only technical details depend on the type of motion considered: In the case of rotational wavepackets (superpositions of J, J and M, or J,M and K eigenstates) the matrix elements of the interaction are analytically expressible, see Eqs. (58) and (70). In the case of electronic wavepakets (wavepackets of Rydberg states) quasi-classical solutions are useful. [See. e.g., L.A. Bureeva, Sov.Phys. Astronomy 12, 962 (1969); V.A. Davidkin and B.A. Zon, Opt.Spectrosc.(USSR) 51, 13 (1981); N.B. Delone, S.P. Goreslavsky and V.P. Krainov, J.Phys.B 27, 4403 (1994).] In the case of vibrational wavepackets, it is often convenient to expand the Hamiltonian on a grid and propagate the wavepacket in time by application of the evolution operator,

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Expansion in terms of the eigenstates of H 0 is advantageous if the Hamiltonian can be stored in the core memory. In cases where the field-free Hamiltonian is too complex to that end, an expansion in terms of an approximate time-independent Hamiltonian is typically more economical. A good example is nonradiative transitions in polyatomic molecules: S 0 ( 1 A g ) laser internal conversion S 1 ( 1 B 2u ) S 2 ( 1 B 3u ) Phys.Rev.Lett. 93, (2004) “An Optimal Control Approach to Suppression of Radiationless Transitions ” Phys.Rev.Lett. 89, (2002) “ Theory of Time-Resolved Photoelectron Imaging. Nonperturbative Calculation for an Internally Converting Polyatomic Molecule” Fig. 8

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III-2. Electronic Wavepackets In a classical world the atom can be thought of as a point electron moving in an orbit around the nucleus as a planet orbits the sun: One uses the term “orbits” for quantum mechanical H-like wavefunctions but here it means something qualitatively different: Fig. 9

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The classical picture of an electron orbiting a nucleus following Kepler’s laws for planetary systems is wrong but sufficiently attractive that tremendous effort has been devoted to making quantum states that will approach it. An electronic wavepacket can do the job (in principle). We require a state that is localized (within Heisenberg’s restrictions) and nonstationary: A classical electron An idealized wavepacket The most common way of producing an electronic wavepacket with the desired properties is to excite an atom from the ground state into a superposition of high Rydberg states with a laser pulse of few to few tens of picoseconds duration: [Rydberg, 1889 ] Fig. 10

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The simplest electronic wavepacket is a radial wavepacket, basically a radial shell of probability which breathes in and out with exactly the period of a classical electron of the same energy: A radial wavepacket An ensemble of classical electrons moving together A quantum mechanical simulation At the instance the wavepacket is formed After ¼ of the classical orbital period After ½ of the classical orbital period Time delay (ps) Ionization signal Experimental results The wavepacket is ionized periodically, when it revisits the core Yeazell et al., Phys.Rev. A 40, 5040 (1989) that’s not quite it though classical period Fig. 11

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III-3. Vibrational Wavepackets To make them in the lab short pulses are necessary: Consider the B state of I 2. The vibrational level spacing around v =15-20 is about 69 cm -1. The bandwidth of a 1 ns pulse is ca cm -1. That of a 20 fs pulse is ca. 700 cm -1. Fig. 13

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In the harmonic limit, If all coefficients are equal (c v = c for all v ) and v is large with respect to unity, (87) (88)

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In this approximation the distance traveled in a round trip is then, The center of the wavepacket oscillates sinusoidally at the harmonic frequency, much like a classical particle: Consider I 2 molecules, excited from the ground (X) into the excited (B) state with a 565 nm, 20 fs long pulse, where about 10 levels are coherently excited, m=1.05 x g, and  =1.3 x Hz. In the approximation of a harmonic potential and equal coefficients, c v = c for all v, c 2 =0.1, yielding a distance of about 1.06Å. x V (89) Fig. 14

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Time in picoseconds measured with respect to the center of the pump pulse 340 fs Fig. 16

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Time in picoseconds, measured with respect to the center of the pump pulse Wavepacket dephasing: Wavepacket revivals: At multiples of T rev =2  /  the t = 0 wavepacket is reconstructed. Fractional revivals appear at shorter times (the half- revival, corresponding to T rev /2 ≈ 18 ps for I 2, is seen in Fig. 17) As t becomes non-negligible with respect to 1/  the wavepacket components step out of phase. Fig. 17

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Whereas electronic and vibrational wavepackets have been around for several decades, rotational wavepackets are a newer concept. They have properties in common with the electronic and vibrational analogs but: Rotational spectra are dense (for I 2, for instance, the vibrational level spacing is cm -1 whereas the rotational level spacing is 2JB, where B is cm -1 ) Rotational spectra are highly nonlinear Electric dipole selection rules forbid the excitation of rotational wavepackets in the weak- field (one-photon) limit The c j vs j distribution is a real arithmetic, simple (essentially Gaussian) function for most types of wavepackets (e.g., j=n, v), whereas for the rotational case the c j exhibit an asymmetric distribution with a j-dependent phase The revival pattern of rotational wavepackets is radically different from the common case III-4. Rotational Wavepackets J.Chem.Phys. 103, 7887 (1995)